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 \chapter*{Introduction}
 \markboth{\quad\ Introduction\hfill}{}
 \addcontentsline{toc}{chapter}{\numberline{\thechapter}Introduction}

 \section{What is geometry about?}
 \markright{\hfill \thesection\ What is geometry about?\quad}
 Geometry ``measuring the world'' attempts to describe and understand space
 around us and all that is in it. It is the central activity and main
 driving force in many branches of math and physics, and offers a whole
 range of views on the nature and meaning of the universe. This book treats
 geometry in a wide context, including a wealth of relations with
 surrounding areas of math and other aspects of human experience.

 Any discussion of geometry involves tension between the twin ideals of
 intuition and precision. {\em Descriptive} or {\em synthetic} geometry
 takes as its starting point our ideas and experience of the observed world,
 and treats geometric objects such as lines and shapes as objects in their
 own right. For example, a line could be the path of a light ray in space;
 you can envisage comparing line segments or angles by ``moving'' one over
 another, thus giving rise to notions of ``congruent'' figures, equal
 lengths, or equal angles that are independent of any quantitative
 measurement. If $A,B,C$ are points along a line segment, what it means for
 $B$ to be {\em between} $A$ and $C$ is an idea hard-wired into our
 consciousness. While descriptive geometry is intuitive and natural, and can
 be made mathematically rigorous (and, of course, Euclidean geometry was
 studied in these terms for more than two millennia,
 compare~\ref{sec!history}), this is not my main
 approach in this book.

 My treatment centres rather on {\em coordinate} geo\-metry. This uses
 Descartes' idea (1637) of measuring distances to view points of space and
 geometric quantities in terms of numbers, with respect to a fixed origin,
 using intuitive ideas such as ``a bit to the right'' or ``a long way up''
 and using them quantitatively in a systematic and precise way. In other
 words, I set up the $(x,y)$-plane $\R^2$, the $(x,y,z)$-space $\R^3$ or
 whatever I need, and use it as a mathematical model of the plane (space,
 etc.), for the purposes of calculations. For example, to plan the layout of
 a car park, I might map it onto a sheet of paper or a computer screen,
 pretending that pairs $(x,y)$ of real numbers correspond to points of the
 surface of the earth, at least in the limited region for which I have
 planning permission. Geometric constructions, such as drawing an even
 rectangular grid or planning the position of the ticket machines to ensure
 the maximum aggravation to customers, are easier to make in the model than
 in real life. We admit possible
 \begin{figure}[h]
 \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
 \vskip4cm
 \centering
 \caption{A coordinate model of space}
 \label{coor_model}
 \end{figure}
drawbacks of our model, but its use divides any problem into calculations
within the model, and considerations of how well it reflects the practical
world.

{\em Topology} is the youngster of the geometry family. Compared to its
venerable predecessors, it really only got going in the 20th century. It
dispenses with practically all the familiar quantities central to other
branches of geo\-metry, such as distance, angles, cross-ratios, and so on.
If you are tempted to the conclusion that there is not much left for
topology to study, think again. Whether two loops of string are linked or
not does not depend on length or shape or perspective; if that seems too
simple to be a serious object of study, what about the linking or knotting
of strands of DNA, or planning the over- and undercrossings on a microchip?
The higher dimensional analogues of disconnecting or knotting are highly
nontrivial and not at all intuitive to denizens of the lower dimensions such
as ourselves, and cannot be discussed without formal apparatus. My treatment
of topology runs briefly through abstract {\em point-set topology}, a fairly
harmless generalisation of the notion of continuity from a first course on
analysis and metric spaces. However, my main interest is in topology as {\em
rubber-sheet geometry}, dealing with manifestly geometric ideas such as
closed curves, spheres, the torus, the M\"obius strip and the Klein bottle.

\section[Change of coordinates, motions, group theory and the \\ Erlangen
program]{Change of coordinates, motions, group \\ theory and the Erlangen
program}
\markright{\hfill \thesection\ Coordinates, motions, groups\quad}

Descartes' idea to use numbers to describe points in space involves the
choice of a {\em coordinate system\/}\index{frame} or {\em coordinate
frame\/}: an origin, together with axes and units of length along the axes.
A recurring theme of all the different geometries in this book is the
question of what a coordinate frame is, and what I can get out of it. While
coordinates provide a convenient framework to discuss points, lines, and so
on, it is a basic requirement that any meaningful statement in geometry is
{\em independent of the choice of coordinates}. That is, coordinate frames
are a humble technical aid in determining the truth, and are not allowed the
dignity of having their own meaning.

Changing from one coordinate frame to another can be viewed as a {\em
transformation\/} or {\em motion\/}\index{motion}: I can use a motion of
space to align the origin and coordinate axes of two coordinate systems. A
statement that remains true under any such motion is independent of the
choice of coordinates. Felix Klein's 1872 {\em Erlangen program} formalises
this relation between geometric properties and changes of coordinates by
defining geo\-metry to be the study of properties invariant under allowed
coordinate changes, that is, invariant under a {\em group of
transformations}. This approach is closely related to the point of view of
special relativity in theoretical physics (Einstein, 1905), which insists
that the laws of physics must be invariant under Lorentz transformations.

This course discusses several different geometries: in some case the spaces
themselves are different (for example, the sphere and the plane), but in
others the difference is purely in the conventions I make about coordinate
changes. Metric geometries such as Euclidean and hyper\-bolic non-Euclidean
geometry include the notion of distance between two points and angle between
two lines. The allowed transformations are rigid motions (iso\-metries or
congruences) of Euclidean or hyperbolic space. Affine and projective
geo\-metries consider properties such as collinearity of points, and the
typical group is the general linear group $\GL(n)$, the group of invertible
$n\times n$ matrixes. Projective geo\-metry presents an interesting paradox:
while its mathematical treatment involves what may seem to be quite arcane
calculations, your brain has a sight driver that carries out projective
transformations by the thousand every time you recognise an object in
perspective, and does so unconsciously and practically instantaneously.

The sets of transformations that appear in topology, for example the set of
all continuous one-to-one maps of the interval $[0,1]$ to itself, or the
same thing for the circle $S^1$ or the sphere $S^2$, are of course too big
for us to study by analogy with transformation groups such as $\GL(n)$ or
the Euclidean group, whose elements depend on finitely many parameters. In
the spirit of the Erlangen program, properties of spaces that remain
invariant under such a huge set of equivalences must be correspondingly
coarse. I treat a few basic topological properties such as compactness,
connectedness, winding number and simple connectedness that appear in many
different areas of analysis and geometry. I use these simple ideas to
motivate the central problem of topology: how to distinguish between
topologically different spaces? At a more advanced level, topology has
developed systematic invariants that apply to this problem, notably the
fundamental group and homology groups. These are invariants of spaces that
are the same for topologically equivalent spaces. Thus if you can calculate
one of these invariants for two spaces (for example, a disc and a punctured
disc) and prove that the answers are different, then the spaces are
certainly not topologically equivalent. You may want to take subsequent
courses in topology to become a real expert, and this course should serve as
a useful guide in this.

 \section{Geometry in applications}
 \markright{\hfill\thesection\ Geometry in applications\quad}

Although this is book is primarily intended for use in a math course, and
the topics are oriented towards the theoretical foundations of geometry, I
must stress that the math ideas discussed here are applicable in different
ways, basic or sophisticated, as stated or with extra development, on their
own or in combination with other disciplines, Euclidean or non-Euclidean,
metric or topological, to a huge variety of scientific and technological
problems in the modern world. I discuss in Chapter~\ref{ch!lie} the quantum
mechanical description of the electron that illustrates a fundamental
application of the ideas of group theory and topology to the physics of
elementary particles. To move away from basic to more applied science, let
me mention a few examples from technology. The typesetting and page layout
software now used throughout the newspaper and publishing industry, as well
as in the computer rooms of most university departments, can obviously not
exist without a knowledge of basic coordinate geometry: even a primary
instruction such as `place letter $A$ or box $B$, scaled by such-and-such a
factor, slanted at such-and-such an angle, at such-and-such a point on the
page' involves affine transformations.\index{affine!transformation} Within
the same industry, computer typefaces themselves are designed using Bezier
curves. The geometry used in robotics is more sophisticated. The
technological aim is, say, to get a robot arm holding a spanner into the
right position and orientation, by adjusting some parameters, say, angles at
joints or lengths of rods. This translates in a fairly obvious way into the
geometric problem of parametrising a piece of the Euclidean
group;\index{Euclidean!group $\protect\Eucl(n)$} but the solution or
approximate solution of this problem is hard, involving the topology and
analysis of manifolds, algebraic geometry and singularity theory. The
computer processing of camera images, whose applications include missile
guidance systems, depends among other things on projective transformations
(I say this for the benefit of students looking for a career truly worthy of
their talents and education). Although scarcely having the same nobility of
purpose, similar techniques apply in ultrasonic scan used in antenatal
clinics; here the geometric problem is to map the variations in density in a
$3$-dimensional medium onto a $2$-dimensional computer screen using
ultrasonic radar, from which the human eye can easily make out salient
features. By a curious coincidence, 3 hours before the senior author gave
the first lecture of this course in January 1989, I was at the maternity
clinic of Walsgrave hospital Coventry looking at just such an image of a
16-week old fetus, now my third daughter Murasaki.

\section{About this book}
 \markright{\hfill \thesection\ About this book\quad}

\subsection*{Who the book is for}

This book is intended for the early years of study of an undergraduate math
course. For the most part, it is based on a second year module taught at
Warwick over many years, a module that is also taken by first and third year
math students, and by students from the math/physics course. You will find
the book accessible if you are familiar with most of the following, which is
standard material in first and second year math courses.

\paragraph{Coordinate geometry:}
How to express lines and circles in $\R^2$ in terms of coordinates, and
calculate their points of intersection; some idea of how to do the same in
$\R^3$ and maybe $\R^n$ may also be helpful.

\paragraph{Linear algebra:}
Vector spaces and linear maps over $\R$ and $\C$, bases and matrixes, change
of bases, eigenvalues and eigenvectors. This is the only major piece of math
that I take for granted. The examples and exercises make occasional
reference to vector spaces over fields other than $\R$ or $\C$ (such as
finite fields), but you can always omit these bits if they make you
uncomfortable.

\paragraph{Multilinear algebra:}
Bilinear and quadratic forms, and how to express them in matrix terms; also
Hermitian forms. I summarise all the necessary background material in
Appendix~\ref{app!q}.

\paragraph{Metric spaces:}
Some prior familiarity with the first ideas of a metric space course would
not do any harm, but this is elementary material, and
Appen\-dix~\ref{app!metric} contains all that you need to know.

\paragraph{Group theory:}
I have gone to some trouble to develop from first principles all the group
theory that I need, with the intention that my book can serve as a first
introduction to transformation groups and the notions of abstract group
theory if you have never seen these. However, if you already have some idea
of basic things such as composition laws, subgroups, cosets and the
symmetric group, these will come in handy as motivation. (If you prefer to
see a conventional introduction to group theory, there are any number of
textbooks, for example Green~\cite{Gr} or Ledermann~\cite{Le}. If you intend
to study group theory beyond the introductory stage, I strongly recommend
Artin~\cite{Ar} or Segal~\cite{Se}.) My ideological slant on this issue is
discussed in more detail in \ref{sec!gp_thy}.

\subsection*{How to use the book}

Although the thousands queuing impatiently at super\-markets and airports to
get their hands on a copy of this book for vacation reading was strong
motivation for me in writing it, experience suggests the harsher view of
reality: at least some of my readers may benefit from coercion in the form
of an organised lecture course.

Experience from teaching at Warwick shows that Chapters~1--6 make a
reasonably paced 30 hour second year lecture course. Some more meat could be
added to subjects that the lecturer or students find interesting; reflection
groups following Coxeter~\cite{Cx}, Chapter~4 would be one good candidate.
Topics from Chapters~7--8 or the further topics of Section~\ref{sec!titles}
could then profitably be assigned to students as essay or project material.
An alternative course oriented towards group theory could start with affine
and Euclidean geometry and some elements of topology (maybe as a refresher),
and concentrate on Chapters~3, 6 and~8, possibly concluding with some
material from Segal~\cite{Se}. This would provide motivation and techniques
to study matrix groups from a geometric point of view, one often ignored in
current texts.

\subsection*{The author's identity crisis}

I want the book to be as informal as possible in style. To this end, I
always refer to the student as ``you'', which has the additional advantage
that it is independent of your gender and number. I also refer to myself by
the first person singular, despite the fact that there are two of me. Each
of me has lectured the material many times, and am used to taking personal
responsibility for the truth of my assertions. My model is van der Waerden's
style, who always wrote the crisp ``Ich behaupte \dots'' (often when
describing results he learned from Emmy Noether or Emil Artin's lectures). I
leave you to imagine the speaker as your ideal teacher, be it a bearded
patriarch or a fresh-faced bespectacled young Central European intellectual.

\subsection*{Acknowledgements}

A second year course with the title `Geometry' or `Geometry and topology'
has been given at Warwick since the 1960s. It goes without saying that my
choice of material, and sometimes the material itself, is taken in part from
the experience of colleagues, including John Jones, Colin Rourke, Brian
Sanderson; David Epstein has also provided some valuable material, notably
in the chapter on hyperbolic geometry. I have also copied material
consciously or unconsciously from several of the textbooks recommended for
the course, in particular Coxeter~\cite{Cx}, Rees~\cite{Rees}, Nikulin and
Shafarevich~\cite{NSh} and Feynman~\cite{Fn}. I owe special thanks to Katrin
Wendland, the most recent lecturer of the Warwick course MA243 Geometry, who
has provided a detailed criticism of my text, thereby saving me from a
variety of embarrassments.

\subsection*{Disclaimer}
 \begin{quote}
Wen solche Lehren nicht erfreun,\\ Verdienet nicht ein Mensch zu sein. \\[6pt]
\hbox{\qquad} From Sarastro's aria, The Magic Flute, II.3.
 \end{quote}
This is an optional course. If you don't like my teaching, please
deregister before the deadline.
